Question

A radioactive substance decays exponentially. A scientist begins with 170 milligrams of a radioactive substance. After 27 hours, 85 mg of the substance remains. How many milligrams will remain after 49 hours?

Answer 1

`\bf A=P(1+r)^t\qquad \begin{cases} P=\textit{starting amount}\\ r=\textit{rate of change}\\ A=\textit{current amount}\\ t=\textit{elapsed time}\\ --------\\ P=170\\ t=27\\ A=85 \end{cases}\implies 85=170(1+r)^(27) \\\\\\ \cfrac{85}{170}=(1+r)^(27)\implies \sqrt[27]{\cfrac{1}{2}}=1+r\implies \boxed{\cfrac{1}{\sqrt[27]{2}}-1=r}\\\\ -----------------------------\\\\ A=170\left( 1+\cfrac{1}{\sqrt[27]{2}}-1 \right)^t \iff A=170\left( \cfrac{1}{\sqrt[27]{2}} \right)^t`

how many mgs after 49hrs? well, set t = 49, to see how much A is there